For the remaining comparisons, we show only the results using the PID controller.
If we plot on a log-log scale, the “work” done by the method, which equates to the number of function evaluations calls and is measured as the number of stages (\(s\) ) of the method multiplied by the total number of time-steps need to successfully integrate to the final time, versus the maximal global error (“precision”) we obtain the a work-precision diagram. In what follows next, we describe the test problems and provide the associated work-precision results; we use WENO5 [1]}, [2]}, a fifth-order accurate method for the spatial discretization of the PDEs.
